1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
|
program sndrv4
c
c Simple program to illustrate the idea of reverse communication
c in shift-invert mode for a generalized nonsymmetric eigenvalue
c problem.
c
c We implement example four of ex-nonsym.doc in DOCUMENTS directory
c
c\Example-4
c ... Suppose we want to solve A*x = lambda*B*x in inverse mode,
c where A and B are derived from the finite element discretization
c of the 1-dimensional convection-diffusion operator
c (d^2u / dx^2) + rho*(du/dx)
c on the interval [0,1] with zero Dirichlet boundary condition
c using linear elements.
c
c ... The shift sigma is a real number.
c
c ... OP = inv[A-SIGMA*M]*M and B = M.
c
c ... Use mode 3 of SNAUPD.
c
c\BeginLib
c
c\Routines called:
c snaupd ARPACK reverse communication interface routine.
c sneupd ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c sgttrf LAPACK tridiagonal factorization routine.
c sgttrs LAPACK tridiagonal linear system solve routine.
c slapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c saxpy Level 1 BLAS that computes y <- alpha*x+y.
c scopy Level 1 BLAS that copies one vector to another.
c sdot Level 1 BLAS that computes the dot product of two vectors.
c snrm2 Level 1 BLAS that computes the norm of a vector.
c av Matrix vector multiplication routine that computes A*x.
c mv Matrix vector multiplication routine that computes M*x.
c
c\Author
c Richard Lehoucq
c Danny Sorensen
c Chao Yang
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\SCCS Information: @(#)
c FILE: ndrv4.F SID: 2.5 DATE OF SID: 10/17/00 RELEASE: 2
c
c\Remarks
c 1. None
c
c\EndLib
c-----------------------------------------------------------------------
c
c %-----------------------------%
c | Define leading dimensions |
c | for all arrays. |
c | MAXN: Maximum dimension |
c | of the A allowed. |
c | MAXNEV: Maximum NEV allowed |
c | MAXNCV: Maximum NCV allowed |
c %-----------------------------%
c
integer maxn, maxnev, maxncv, ldv
parameter (maxn=256, maxnev=10, maxncv=25,
& ldv=maxn )
c
c %--------------%
c | Local Arrays |
c %--------------%
c
integer iparam(11), ipntr(14), ipiv(maxn)
logical select(maxncv)
Real
& ax(maxn), mx(maxn), d(maxncv,3), resid(maxn),
& v(ldv,maxncv), workd(3*maxn), workev(3*maxncv),
& workl(3*maxncv*maxncv+6*maxncv),
& dd(maxn), dl(maxn), du(maxn),
& du2(maxn)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character bmat*1, which*2
integer ido, n, nev, ncv, lworkl, info, ierr, j,
& nconv, maxitr, ishfts, mode
Real
& tol, h, s,
& sigmar, sigmai, s1, s2, s3
logical first, rvec
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Real
& sdot, snrm2, slapy2
external sdot, snrm2, slapy2, sgttrf, sgttrs
c
c %--------------------%
c | Intrinsic function |
c %--------------------%
c
intrinsic abs
c
c %------------%
c | Parameters |
c %------------%
c
Real
& one, zero, two, six, rho
common /convct/ rho
parameter (one = 1.0E+0, zero = 0.0E+0,
& two = 2.0E+0, six = 6.0E+0)
c
c %-----------------------%
c | Executable statements |
c %-----------------------%
c
c %----------------------------------------------------%
c | The number N is the dimension of the matrix. A |
c | generalized eigenvalue problem is solved (BMAT = |
c | 'G'). NEV is the number of eigenvalues (closest |
c | to SIGMAR) to be approximated. Since the |
c | shift-invert mode is used, WHICH is set to 'LM'. |
c | The user can modify NEV, NCV, SIGMAR to solve |
c | problems of different sizes, and to get different |
c | parts of the spectrum. However, The following |
c | conditions must be satisfied: |
c | N <= MAXN, |
c | NEV <= MAXNEV, |
c | NEV + 2 <= NCV <= MAXNCV |
c %----------------------------------------------------%
c
n = 100
nev = 4
ncv = 10
if ( n .gt. maxn ) then
print *, ' ERROR with _NDRV4: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV4: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV4: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'G'
which = 'LM'
sigmar = one
sigmai = zero
c
c %--------------------------------------------------%
c | Construct C = A - SIGMA*M in real arithmetic, |
c | and factor C in real arithmetic (using LAPACK |
c | subroutine sgttrf). The matrix A is chosen to be |
c | the tridiagonal matrix derived from the standard |
c | central difference discretization of the 1-d |
c | convection-diffusion operator u" + rho*u' on the |
c | interval [0, 1] with zero Dirichlet boundary |
c | condition. The matrix M is the mass matrix |
c | formed by using piecewise linear elements on |
c | [0,1]. |
c %--------------------------------------------------%
c
rho = 1.0E+1
h = one / real(n+1)
s = rho / two
c
s1 = -one/h - s - sigmar*h/six
s2 = two/h - 4.0E+0*sigmar*h/six
s3 = -one/h + s - sigmar*h/six
c
do 10 j = 1, n-1
dl(j) = s1
dd(j) = s2
du(j) = s3
10 continue
dd(n) = s2
c
call sgttrf(n, dl, dd, du, du2, ipiv, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrf in _NDRV4.'
print*, ' '
go to 9000
end if
c
c %-----------------------------------------------------%
c | The work array WORKL is used in SNAUPD as |
c | workspace. Its dimension LWORKL is set as |
c | illustrated below. The parameter TOL determines |
c | the stopping criterion. If TOL<=0, machine |
c | precision is used. The variable IDO is used for |
c | reverse communication, and is initially set to 0. |
c | Setting INFO=0 indicates that a random vector is |
c | generated in SNAUPD to start the Arnoldi iteration. |
c %-----------------------------------------------------%
c
lworkl = 3*ncv**2+6*ncv
tol = zero
ido = 0
info = 0
c
c %---------------------------------------------------%
c | This program uses exact shifts with respect to |
c | the current Hessenberg matrix (IPARAM(1) = 1). |
c | IPARAM(3) specifies the maximum number of Arnoldi |
c | iterations allowed. Mode 3 of SNAUPD is used |
c | (IPARAM(7) = 3). All these options can be |
c | changed by the user. For details, see the |
c | documentation in SNAUPD. |
c %---------------------------------------------------%
c
ishfts = 1
maxitr = 300
mode = 3
c
iparam(1) = ishfts
iparam(3) = maxitr
iparam(7) = mode
c
c %------------------------------------------%
c | M A I N L O O P(Reverse communication) |
c %------------------------------------------%
c
20 continue
c
c %---------------------------------------------%
c | Repeatedly call the routine SNAUPD and take |
c | actions indicated by parameter IDO until |
c | either convergence is indicated or maxitr |
c | has been exceeded. |
c %---------------------------------------------%
c
call snaupd ( ido, bmat, n, which, nev, tol, resid,
& ncv, v, ldv, iparam, ipntr, workd,
& workl, lworkl, info )
c
if (ido .eq. -1) then
c
c %-------------------------------------------%
c | Perform y <--- OP*x = inv[A-SIGMA*M]*M*x |
c | to force starting vector into the range |
c | of OP. The user should supply his/her |
c | own matrix vector multiplication routine |
c | and a linear system solver. The matrix |
c | vector multiplication routine should take |
c | workd(ipntr(1)) as the input. The final |
c | result should be returned to |
c | workd(ipntr(2)). |
c %-------------------------------------------%
c
call mv (n, workd(ipntr(1)), workd(ipntr(2)))
call sgttrs('N', n, 1, dl, dd, du, du2, ipiv,
& workd(ipntr(2)), n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrs in _NDRV4.'
print*, ' '
go to 9000
end if
c
c %-----------------------------------------%
c | L O O P B A C K to call SNAUPD again. |
c %-----------------------------------------%
c
go to 20
c
else if ( ido .eq. 1) then
c
c %-----------------------------------------%
c | Perform y <-- OP*x = inv[A-sigma*M]*M*x |
c | M*x has been saved in workd(ipntr(3)). |
c | The user only need the linear system |
c | solver here that takes workd(ipntr(3)) |
c | as input, and returns the result to |
c | workd(ipntr(2)). |
c %-----------------------------------------%
c
call scopy( n, workd(ipntr(3)), 1, workd(ipntr(2)), 1)
call sgttrs ('N', n, 1, dl, dd, du, du2, ipiv,
& workd(ipntr(2)), n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrs in _NDRV4.'
print*, ' '
go to 9000
end if
c
c %-----------------------------------------%
c | L O O P B A C K to call SNAUPD again. |
c %-----------------------------------------%
c
go to 20
c
else if ( ido .eq. 2) then
c
c %---------------------------------------------%
c | Perform y <--- M*x |
c | Need matrix vector multiplication routine |
c | here that takes workd(ipntr(1)) as input |
c | and returns the result to workd(ipntr(2)). |
c %---------------------------------------------%
c
call mv (n, workd(ipntr(1)), workd(ipntr(2)))
c
c %-----------------------------------------%
c | L O O P B A C K to call SNAUPD again. |
c %-----------------------------------------%
c
go to 20
c
end if
c
c
c %-----------------------------------------%
c | Either we have convergence, or there is |
c | an error. |
c %-----------------------------------------%
c
if ( info .lt. 0 ) then
c
c %--------------------------%
c | Error message, check the |
c | documentation in SNAUPD. |
c %--------------------------%
c
print *, ' '
print *, ' Error with _naupd, info = ', info
print *, ' Check the documentation in _naupd.'
print *, ' '
c
else
c
c %-------------------------------------------%
c | No fatal errors occurred. |
c | Post-Process using SNEUPD. |
c | |
c | Computed eigenvalues may be extracted. |
c | |
c | Eigenvectors may also be computed now if |
c | desired. (indicated by rvec = .true.) |
c %-------------------------------------------%
c
rvec = .true.
call sneupd ( rvec, 'A', select, d, d(1,2), v, ldv,
& sigmar, sigmai, workev, bmat, n, which, nev, tol,
& resid, ncv, v, ldv, iparam, ipntr, workd,
& workl, lworkl, ierr )
c
c %-----------------------------------------------%
c | The real part of the eigenvalue is returned |
c | in the first column of the two dimensional |
c | array D, and the IMAGINARY part is returned |
c | in the second column of D. The corresponding |
c | eigenvectors are returned in the first NEV |
c | columns of the two dimensional array V if |
c | requested. Otherwise, an orthogonal basis |
c | for the invariant subspace corresponding to |
c | the eigenvalues in D is returned in V. |
c %-----------------------------------------------%
c
if ( ierr .ne. 0) then
c
c %------------------------------------%
c | Error condition: |
c | Check the documentation of SNEUPD. |
c %------------------------------------%
c
print *, ' '
print *, ' Error with _neupd, info = ', ierr
print *, ' Check the documentation of _neupd. '
print *, ' '
c
else
c
first = .true.
nconv = iparam(5)
do 30 j=1, nconv
c
c %---------------------------%
c | Compute the residual norm |
c | |
c | || A*x - lambda*x || |
c | |
c | for the NCONV accurately |
c | computed eigenvalues and |
c | eigenvectors. (iparam(5) |
c | indicates how many are |
c | accurate to the requested |
c | tolerance) |
c %---------------------------%
c
if (d(j,2) .eq. zero) then
c
c %--------------------%
c | Ritz value is real |
c %--------------------%
c
call av(n, v(1,j), ax)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
d(j,3) = snrm2(n, ax, 1)
d(j,3) = d(j,3) / abs(d(j,1))
c
else if (first) then
c
c %------------------------%
c | Ritz value is complex. |
c | Residual of one Ritz |
c | value of the conjugate |
c | pair is computed. |
c %------------------------%
c
call av(n, v(1,j), ax)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j+1), mx)
call saxpy(n, d(j,2), mx, 1, ax, 1)
d(j,3) = snrm2(n, ax, 1)
call av(n, v(1,j+1), ax)
call mv(n, v(1,j+1), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,2), mx, 1, ax, 1)
d(j,3) = slapy2( d(j,3), snrm2(n, ax, 1) )
d(j,3) = d(j,3) / slapy2(d(j,1),d(j,2))
d(j+1,3) = d(j,3)
first = .false.
else
first = .true.
end if
c
30 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call smout(6, nconv, 3, d, maxncv, -6,
& 'Ritz values (Real,Imag) and relative residuals')
c
end if
c
c %-------------------------------------------%
c | Print additional convergence information. |
c %-------------------------------------------%
c
if ( info .eq. 1) then
print *, ' '
print *, ' Maximum number of iterations reached.'
print *, ' '
else if ( info .eq. 3) then
print *, ' '
print *, ' No shifts could be applied during implicit',
& ' Arnoldi update, try increasing NCV.'
print *, ' '
end if
c
print *, ' '
print *, ' _NDRV4 '
print *, ' ====== '
print *, ' '
print *, ' Size of the matrix is ', n
print *, ' The number of Ritz values requested is ', nev
print *, ' The number of Arnoldi vectors generated',
& ' (NCV) is ', ncv
print *, ' What portion of the spectrum: ', which
print *, ' The number of converged Ritz values is ',
& nconv
print *, ' The number of Implicit Arnoldi update',
& ' iterations taken is ', iparam(3)
print *, ' The number of OP*x is ', iparam(9)
print *, ' The convergence criterion is ', tol
print *, ' '
c
end if
c
c %---------------------------%
c | Done with program sndrv4. |
c %---------------------------%
c
9000 continue
c
end
c
c==========================================================================
c
c matrix vector multiplication subroutine
c
subroutine mv (n, v, w)
integer n, j
Real
& v(n), w(n), one, four, six, h
parameter (one = 1.0E+0, four = 4.0E+0, six = 6.0E+0)
c
c Compute the matrix vector multiplication y<---M*x
c where M is mass matrix formed by using piecewise linear elements
c on [0,1].
c
w(1) = ( four*v(1) + one*v(2) ) / six
do 10 j = 2,n-1
w(j) = ( one*v(j-1) + four*v(j) + one*v(j+1) ) / six
10 continue
w(n) = ( one*v(n-1) + four*v(n) ) / six
c
h = one / real(n+1)
call sscal(n, h, w, 1)
return
end
c------------------------------------------------------------------
subroutine av (n, v, w)
integer n, j
Real
& v(n), w(n), one, two, dd, dl, du, s, h, rho
common /convct/ rho
parameter (one = 1.0E+0, two = 2.0E+0)
c
c Compute the matrix vector multiplication y<---A*x
c where A is obtained from the finite element discretization of the
c 1-dimensional convection diffusion operator
c d^u/dx^2 + rho*(du/dx)
c on the interval [0,1] with zero Dirichlet boundary condition
c using linear elements.
c This routine is only used in residual calculation.
c
h = one / real(n+1)
s = rho / two
dd = two / h
dl = -one/h - s
du = -one/h + s
c
w(1) = dd*v(1) + du*v(2)
do 10 j = 2,n-1
w(j) = dl*v(j-1) + dd*v(j) + du*v(j+1)
10 continue
w(n) = dl*v(n-1) + dd*v(n)
return
end
|
